As a consequence of the discussion in the preceding sections we obtain the equivalence of étale cohomology of the spectrum of a field with Galois cohomology.
Lemma 59.59.1. Let S = \mathop{\mathrm{Spec}}(K) with K a field. Let \overline{s} be a geometric point of S. Let G = \text{Gal}_{\kappa (s)} denote the absolute Galois group. The stalk functor induces an equivalence of categories
Proof. In Theorem 59.56.3 we have seen the equivalence between sheaves of sets and G-sets. The current lemma follows formally from this as an abelian sheaf is just a sheaf of sets endowed with a commutative group law, and a G-module is just a G-set endowed with a commutative group law. \square
Lemma 59.59.2. Notation and assumptions as in Lemma 59.59.1. Let \mathcal{F} be an abelian sheaf on \mathop{\mathrm{Spec}}(K)_{\acute{e}tale} which corresponds to the G-module M. Then
in D(\textit{Ab}) we have a canonical isomorphism R\Gamma (S, \mathcal{F}) = R\Gamma _ G(M),
H_{\acute{e}tale}^0(S, \mathcal{F}) = M^ G, and
H_{\acute{e}tale}^ q(S, \mathcal{F}) = H^ q(G, M).
Example 59.59.3. Sheaves on \mathop{\mathrm{Spec}}(K)_{\acute{e}tale}. Let G = \text{Gal}(K^{sep}/K) be the absolute Galois group of K.
The constant sheaf \underline{\mathbf{Z}/n\mathbf{Z}} corresponds to the module \mathbf{Z}/n\mathbf{Z} with trivial G-action,
the sheaf \mathbf{G}_ m|_{\mathop{\mathrm{Spec}}(K)_{\acute{e}tale}} corresponds to (K^{sep})^* with its G-action,
the sheaf \mathbf{G}_ a|_{\mathop{\mathrm{Spec}}(K^{sep})} corresponds to (K^{sep}, +) with its G-action, and
the sheaf \mu _ n|_{\mathop{\mathrm{Spec}}(K^{sep})} corresponds to \mu _ n(K^{sep}) with its G-action.
By Remark 59.23.4 and Theorem 59.24.1 we have the following identifications for cohomology groups:
Also, for any quasi-coherent sheaf \mathcal{F} on S_{\acute{e}tale} we have
see Theorem 59.22.4. In particular, this gives the following sequence of equalities
which is none other than Hilbert's 90 theorem. Similarly, for i \geq 1,
where the K^{sep} indicates K^{sep} as a Galois module with addition as group law. In this way we may consider the work we have done so far as a complicated way of computing Galois cohomology groups.
The following result is a curiosity and should be skipped on a first reading.
Lemma 59.59.4. Let R be a local ring of dimension 0. Let S = \mathop{\mathrm{Spec}}(R). Then every \mathcal{O}_ S-module on S_{\acute{e}tale} is quasi-coherent.
Proof. Let \mathcal{F} be an \mathcal{O}_ S-module on S_{\acute{e}tale}. We have to show that \mathcal{F} is determined by the R-module M = \Gamma (S, \mathcal{F}). More precisely, if \pi : X \to S is étale we have to show that \Gamma (X, \mathcal{F}) = \Gamma (X, \pi ^*\widetilde{M}).
Let \mathfrak m \subset R be the maximal ideal and let \kappa be the residue field. By Algebra, Lemma 10.153.10 the local ring R is henselian. If X \to S is étale, then the underlying topological space of X is discrete by Morphisms, Lemma 29.36.7 and hence X is a disjoint union of affine schemes each having one point. Moreover, if X = \mathop{\mathrm{Spec}}(A) is affine and has one point, then R \to A is finite étale by Algebra, Lemma 10.153.5. We have to show that \Gamma (X, \mathcal{F}) = M \otimes _ R A in this case.
The functor A \mapsto A/\mathfrak m A defines an equivalence of the category of finite étale R-algebras with the category of finite separable \kappa -algebras by Algebra, Lemma 10.153.7. Let us first consider the case where A/\mathfrak m A is a Galois extension of \kappa with Galois group G. For each \sigma \in G let \sigma : A \to A denote the corresponding automorphism of A over R. Let N = \Gamma (X, \mathcal{F}). Then \mathop{\mathrm{Spec}}(\sigma ) : X \to X is an automorphism over S and hence pullback by this defines a map \sigma : N \to N which is a \sigma -linear map: \sigma (an) = \sigma (a) \sigma (n) for a \in A and n \in N. We will apply Galois descent to the quasi-coherent module \widetilde{N} on X endowed with the isomorphisms coming from the action on \sigma on N. See Descent, Lemma 35.6.2. This lemma tells us there is an isomorphism N = N^ G \otimes _ R A. On the other hand, it is clear that N^ G = M by the sheaf property for \mathcal{F}. Thus the required isomorphism holds.
The general case (with A local and finite étale over R) is deduced from the Galois case as follows. Choose A \to B finite étale such that B is local with residue field Galois over \kappa . Let G = \text{Aut}(B/R) = \text{Gal}(\kappa _ B/\kappa ). Let H \subset G be the Galois group corresponding to the Galois extension \kappa _ B/\kappa _ A. Then as above one shows that \Gamma (X, \mathcal{F}) = \Gamma (\mathop{\mathrm{Spec}}(B), \mathcal{F})^ H. By the result for Galois extensions (used twice) we get
as desired. \square
Comments (0)